Grothendieck inequalities characterize converses to the polynomial method

• URL: http://arxiv.org/abs/2212.08559v2
• Date: Sat, 30 Dec 2023 12:26:30 GMT
• Title: Grothendieck inequalities characterize converses to the polynomial method
• Authors: Jop Bri\"et, Francisco Escudero Guti\'errez and Sander Gribling
• Abstract summary: A surprising 'converse to the method' of Aaronson et al. (CCC16) shows that any bounded quadratic can be computed exactly by a 1-query up to a universal multiplicative factor related to the Grothendieck constant.
• Score: 1.024113475677323
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.

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